25,347 research outputs found
The inertial dynamics of thin film flow of non-Newtonian fluids
Consider the flow of a thin layer of non-Newtonian fluid over a solid
surface. I model the case of a viscosity that depends nonlinearly on the
shear-rate; power law fluids are an important example, but the analysis here is
for general nonlinear dependence. The modelling allows for large changes in
film thickness provided the changes occur over a large enough lateral length
scale. Modifying the surface boundary condition for tangential stress forms an
accessible base for the analysis where flow with constant shear is a neutral
critical mode, in addition to a mode representing conservation of fluid.
Perturbatively removing the modification then constructs a model for the
coupled dynamics of the fluid depth and the lateral momentum. For example, the
results model the dynamics of gravity currents of non-Newtonian fluids even
when the flow is not very slow
A geometry of information, I: Nerves, posets and differential forms
The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial
Representation: Continuous vs. Discrete'. Spatial representation has two
contrasting but interacting aspects (i) representation of spaces' and (ii)
representation by spaces. In this paper, we will examine two aspects that are
common to both interpretations of the theme, namely nerve constructions and
refinement. Representations change, data changes, spaces change. We will
examine the possibility of a `differential geometry' of spatial representations
of both types, and in the sequel give an algebra of differential forms that has
the potential to handle the dynamical aspect of such a geometry. We will
discuss briefly a conjectured class of spaces, generalising the Cantor set
which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl
seminar portal http://drops.dagstuhl.de/portals/04351
Accurate macroscale modelling of spatial dynamics in multiple dimensions
Developments in dynamical systems theory provides new support for the
macroscale modelling of pdes and other microscale systems such as Lattice
Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically
resolving subgrid microscale dynamics the dynamical systems approach constructs
accurate closures of macroscale discretisations of the microscale system. Here
we specifically explore reaction-diffusion problems in two spatial dimensions
as a prototype of generic systems in multiple dimensions. Our approach unifies
into one the modelling of systems by a type of finite elements, and the
`equation free' macroscale modelling of microscale simulators efficiently
executing only on small patches of the spatial domain. Centre manifold theory
ensures that a closed model exist on the macroscale grid, is emergent, and is
systematically approximated. Dividing space either into overlapping finite
elements or into spatially separated small patches, the specially crafted
inter-element/patch coupling also ensures that the constructed discretisations
are consistent with the microscale system/PDE to as high an order as desired.
Computer algebra handles the considerable algebraic details as seen in the
specific application to the Ginzburg--Landau PDE. However, higher order models
in multiple dimensions require a mixed numerical and algebraic approach that is
also developed. The modelling here may be straightforwardly adapted to a wide
class of reaction-diffusion PDEs and lattice equations in multiple space
dimensions. When applied to patches of microscopic simulations our coupling
conditions promise efficient macroscale simulation.Comment: some figures with 3D interaction when viewed in Acrobat Reader. arXiv
admin note: substantial text overlap with arXiv:0904.085
Model turbulent floods with the Smagorinski large eddy closure
Floods, tides and tsunamis are turbulent, yet conventional models are based
upon depth averaging inviscid irrotational flow equations. We propose to change
the base of such modelling to the Smagorinksi large eddy closure for turbulence
in order to appropriately match the underlying fluid dynamics. Our approach
allows for large changes in fluid depth to cater for extreme inundations. The
key to the analysis underlying the approach is to choose surface and bed
boundary conditions that accommodate a constant turbulent shear as a nearly
neutral mode. Analysis supported by slow manifold theory then constructs a
model for the coupled dynamics of the fluid depth and the mean turbulent
lateral velocity. The model resolves the internal turbulent shear in the flow
and thus may be used in further work to rationally predict erosion and
transport in turbulent floods
Behavioural hybrid process calculus
Process algebra is a theoretical framework for the modelling and analysis of the behaviour of concurrent discrete event systems that has been developed within computer science in past quarter century. It has generated a deeper nderstanding of the nature of concepts such as observable behaviour in the presence of nondeterminism, system composition by interconnection of concurrent component systems, and notions of behavioural equivalence of such systems. It has contributed fundamental concepts such as bisimulation, and has been successfully used in a wide range of problems and practical applications in concurrent systems. We believe that the basic tenets of process algebra are highly compatible with the behavioural approach to dynamical systems. In our contribution we present an extension of classical process algebra that is suitable for the modelling and analysis of continuous and hybrid dynamical systems. It provides a natural framework for the concurrent composition of such systems, and can deal with nondeterministic behaviour that may arise from the occurrence of internal switching events. Standard process algebraic techniques lead to the characterisation of the observable behaviour of such systems as equivalence classes under some suitably adapted notion of bisimulation
Holistic projection of initial conditions onto a finite difference approximation
Modern dynamical systems theory has previously had little to say about finite
difference and finite element approximations of partial differential equations
(Archilla, 1998). However, recently I have shown one way that centre manifold
theory may be used to create and support the spatial discretisation of \pde{}s
such as Burgers' equation (Roberts, 1998a) and the Kuramoto-Sivashinsky
equation (MacKenzie, 2000). In this paper the geometric view of a centre
manifold is used to provide correct initial conditions for numerical
discretisations (Roberts, 1997). The derived projection of initial conditions
follows from the physical processes expressed in the PDEs and so is
appropriately conservative. This rational approach increases the accuracy of
forecasts made with finite difference models.Comment: 8 pages, LaTe
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